Some universal graphs (Q1119949)

If \({\mathcal C}\) is a class of graphs, a graph \(G\in {\mathcal C}\) is called strongly universal in \({\mathcal C}\) if every \(H\in {\mathcal C}\) with \(| H| \leq | G|\) is isomorphic to an induced subgraph of G. The existence of a strongly universal graph of size \(\kappa\) in \({\mathcal C}\) is proved for 1. \(\kappa =\aleph_ 0\), \({\mathcal C}=graphs\) not containing odd circuits of length at most \(2k+1,\) 2. \(\kappa\) arbitrary infinite, \({\mathcal C}=graphs\) without circuits of length at least k, 3. \(\kappa\) arbitrary infinite, \({\mathcal C}=graphs\) without paths of length k.